Dynamical instability in kicked Bose-Einstein condensates

J. Reslen,1 C.E. Creffield,1,2 and T.S. Monteiro11Department of Physics and Astronomy,
University College London, Gower Street, London WC1E 6BT, United Kingdom
2Departamento de Física de Materiales,
Universidad Complutense de Madrid, E-28040, Madrid, Spain

July 12, 2019

Abstract

Bose-Einstein condensates subject to short pulses (‘kicks’)
from standing waves of light represent a nonlinear analogue of the
well-known chaos paradigm, the quantum kicked rotor.
Previous studies of the onset of dynamical instability
(ie exponential proliferation of non-condensate particles)
suggested that the transition to instability might be associated with
a transition to chaos. Here we conclude instead
that instability is due to resonant driving of Bogoliubov
modes.
We investigate the Bogoliubov spectrum for
both the quantum kicked rotor (QKR) and a variant, the
double kicked rotor (QKR-2). We present an analytical model,
valid in the limit of weak impulses which correctly gives the
scaling properties of the resonances and yields good
agreement with mean-field numerics.

pacs:

03.75.Lm,05.45.-a,03.65.Ta,03.75.-b

I Introduction

The production of Bose-Einstein condensates (BECs) in dilute
atomic gases has opened up a new domain for research in quantum dynamics,
since BECs are intrinsically phase-coherent and can be controlled
experimentally to an extremely high degree of precision Stringari ().
An increasingly interesting aspect of the dynamics of BECs is that
they represent a new arena for investigation
of the interaction between nonlinearity and quantum dynamics, including quantum chaos
Shep (); Gardiner (); Garreau (); Duffy (); Zhang (); Zhang2 (); Wimberger (); Adams ().

A BEC subject to periodic short pulses, or kicks, from standing waves of
light represents a nonlinear generalization of the
well-known chaos paradigm, the quantum kicked rotor (QKR).
The QKR has been realized using (non-condensed) cold atoms,
permitting experimental investigation of a range of interesting
chaos phenomena Raizen (). The regime where the kick-period T
is a rational multiple of π has also proved of particular
interest: several studies
have investigated the dynamics here with or without
linearity Darcy (); Fish (); Wimberger (); Rebuzzini (). A number of experimental studies
have also investigated kicked BECs Sadgrove ().
Ensuring dynamical stability of the condensate is thus very
important in studies of its coherent dynamics:
if the condensate is dynamically unstable, numbers of
non-condensate particles grow exponentially. If it is stable, they grow
more slowly (polynomially). More broadly,
the study of diffferent types of instability in static Wu () and driven BECs
Dalfovo () is of much current interest.

Previous work on kicked systemsGardiner (); Zhang (); Zhang2 () considered the onset of
dynamical instability and investigated the relation with
classical chaos. In Gardiner (), the possibility that instability is
related to chaos in the one-body limit was investigated
for the Kicked Harmonic Oscillator.
In Zhang (); Zhang2 () the correlation between
chaos in the mean-field dynamics, rather, and
the onset of dynamical instability, was investigated.
An “instability border”, determined by the
kick strength K and the nonlinearity g was mapped out;
it was then found Zhang2 () that the parameter ranges for this border
corresponds closely to a transition from regular to chaotic motion,
of an effective classical Hamiltonian derived from the mean-field
dynamics. Hence, present understanding of onset of dynamical instability
in kicked BECs suggests that it may somehow be related to a transition to chaos.

In this work, we conclude that a quite different mechanism is primarily responsible
for dynamical instability in the QKR-BEC.
Our key finding is that it is the strong resonant driving of
certain condensate modes by the kicking, which triggers loss of stability
of the condensate. This mechanism is unrelated to the transition to chaos,
but is rather an example of parametric resonance. In another context,
the relationship between parametric
resonance and dynamical instability of a BEC in a time-modulated trap is a topic
of much current theoretical PRTHEO (); Dalfovo () and experimental interest PREXPT ().
But to date, “Bogoliubov spectroscopy” in the analogous time-dependent system, the
δ-kicked BEC, has not been investigated. Our study
shows that the temporally kicked BECs open up many new possibilities in this arena.

We find that in general, for the kicked-BEC,
there is no single stability border: typically, for moderate K,
the condensate restabilizes just above the stability border.
For small K and g the number of non-condensed
atoms Nex(t) grows exponentially only very close to a few, isolated
resonance peaks.
With increasing K and g, the number of resonances which
can be strongly excited by the kicking proliferates and overlaps.
Our calculations show this is associated with generalized
exponential instability; however this regime is, to a
large degree, beyond the scope of our methods.
For lower K and g, though, we introduce a simple perturbative model which
provides the approximate position and
width of the important resonances for both rational and irrational T.

A key finding is that, for the integer values of T/π=m
(where m is integer) values, the focus of the study in Zhang (),
the onset of instability can occur at nonlinearities much lower than those
required to resonantly excite even the very lowest collective mode – a key reason
why the mechanism of parametric resonance may
so far been overlooked in respect of destabilization of kicked BECs.
Our model demonstrates that for this case, resonant excitation involves
two excited modes in addition to the initial ground state mode.
Hence we can explain the position of the critical stability border
found in Zhang (); Poletti ().

We investigate both the usual QKR-BEC
as well as a simple modification, obtained by applying a series of pairs
of closely-spaced opposing kicks (the QKR2-BEC). This modifies substantially
the relative strengths of the resonances, and provides the added novelty
that the lowest modes are excited by an effective imaginary kick-strength.
It is closely related to the double-kicked quantum rotor, investigated
in cold atoms experiments and theory Jones ().
We introduce a simple analytical model based on the properties of the
unperturbed condensate, which gives the distinctive properties and scaling
behavior of the condensate oscillations on and off resonance.

In Section II we introduce briefly the kicked and double-kicked BEC
systems.
In Section III we introduce the time-dependent Bogoliubov method
proposed by Castin and Dum and present numerics for
the growth of non-condensate atoms. In Section IV we introduce a
simple perturbative model, based on the one period time evolution operator
for a kicked BEC. In Section V we show that the simple model and
the time-dependent Bogoliubov numerics give excellent agreement
in the limit of weak kicks. In Section VI we consider the case
T=2π with both numerics and the perturbative model
and show that the instability border found in Zhang (); Zhang2 () is
due to a novel type of compound Bogoliubov resonance.

Ii Kicked BEC systems

As in Zhang (), we consider a BEC confined in a ring-shaped trap of
radius R. We assume that the lateral dimension r
of the trap is much smaller than its circumference, and thus we
are dealing with an effectively 1D system rescalg (). The dynamics of
the condensate wavefunction at temperatures well below the transition
temperature are then governed by the 1D Gross-Pitaevskii (GP)
Hamiltonian with an additional kicking potential:

H=HGP+Kcosθf(t),

(1)

where

HGP=−ℏ22mR2∂2∂θ2+g|ψ(θ,t)|2.

(2)

The short-range interactions between the atoms in the condensate
are described by a mean-field term with strength g=8NtotaSR/r2,
where aS is the s-wave scattering length, and Ntot is
the total number of atoms.
For the QKR-BEC system, f(t)=∑nδ(t−nT), while for the
QKR2-BEC,

f(t)=∑n[δ(t−nT)−δ(t−nT+ϵ)],

(3)

where T is the total period of the driving; ϵ≪T
and thus the second kick nearly cancels the first.

Experimental and theoretical studies of the double-kicked rotor
Jones () have shown that its quantum behavior is
largely determined by an effective kick strength Kϵ=Kϵ,
provided T≫ϵ.
Here we take ϵ=1/25. Hence, while for the QKR-BEC, the value
K=1 represents a relatively large impulse for a kicked BEC,
for a double kicked BEC, K=1 in the numerics below corresponds to
Kϵ=0.04, and represents only a very weak impulse.
The reason for this is the near cancellation of consecutive
kicks in each pair.

This mechanism has certain analogies with the so-called “quantum antiresonance”
investigated in Zhang (): for QKRs kicked at T=2π,
consecutive kicks effectively cancel. This means that
even large values of K≃1 and g>1 represent
only weak driving; for example, the instability border was found by
Zhang () to occur at g≃2 and K=0.8.

Iii time-dependent Bogoliubov method

The number of non-condensed atoms were calculated by making the
usual Bogoliubov approximation, and following the formalism of
Castin and Dum castin (). This adaptation of the Bogoliubov
linearization for time-dependent potentials has been used in
all studies to date of the dynamical stability of kicked
condensates Gardiner (); Zhang (); Poletti (); Rebuzzini ().
The mean number of non-condensed atoms at zero temperature is given by
Nex(t)=∑∞k=1⟨vk(t)|vk(t)⟩,
where the amplitudes (uk,vk) of the Bogoliubov
quasiparticle operators are governed by the coupled equations

iℏddt(ukvk)=(H+gQ|ψ|2QgQψ2Q∗−gQ∗ψ∗2Q−H−gQ∗|ψ|2Q∗)(ukvk).

(4)

In this expression, Q=I−|ψ⟩⟨ψ|
are projection operators that orthogonalize the quasiparticle modes
with respect to the condensate castin ().
We assume that at time t=0, we have a homogeneous condensate
ψ0=1/√2π. Further discussion of the theory is given in
gard ().

The regime of validity of the method is discussed in castin ().
The method is valid in the weakly interacting limit 1≫a3sρ
where ρ is the density. A limit is identified where this condition
is satisfied, if one works with a constant g∝Ntotas; thus the limit
as→0 corresponds to Ntot→∞.
A further requirement is that condensate depletion remains negligible.
This condition fails after a few kicks in exponentially unstable regions.
Here the method is employed only to identify the the parameter range for
the onset of instability.
We cut-off our calculations for Nex>103 (a reasonable threshold for
small depletion in a condensate with Ntot∼105).

In Figs.1 (a) and (b) we show the number of non-condensed atoms,
Nex(t=NT), calculated from the Bogoliubov
equations (4) after N=200.
For small K=0.2, g=1, a single resonance is seen
at T≃10.
For small K, resonances occur whenever the resonance
condition Dalfovo ()ω0+ωl=ωl≈2nπT is satisfied,
where n=1,2,3.. is an integer and ωl is the eigenfrequency
of the l−th collective mode.
For larger K=1, the figure shows that
resonances are extremely dense and overlap with each
other (and we show the behavior in this regime
for T<10). For overlapping resonances, unambiguous
identification of each resonance is no longer possible.
The key point here, however, is that in the stable regions outside the resonances,
Nex remains very small even after prolonged kicking.

Fig.1(b)
shows oscillations of Nex, as a function of time, for weak K=0.2, g=1,
close to the isolated resonance at T≈10. The three possible
regimes of: (non-resonant) weak quasi-periodic oscillations in time;
(near-resonant) slower, large periodic oscillations;
and (resonant) exponential growth are illustrated.
The condensate energy,
E(N)=∫2π0dθψ∗(N)(−12∂2∂θ2+g2|ψ(N)|2)ψ(N)
after N kicks, obtained from the GPE itself, is also shown, for
comparison, in the inset: at resonance, large oscillations are also seen.

Fig.2 shows the corresponding behavior for the double-kicked
BEC, but now as a function of g, keeping T=2, ϵ=1/25 constant
and K=1 or K=5 (hence Kϵ=0.04 or 0.2).
The curve Kϵ=0.04 corresponds to weak impulses and shows two isolated
Bogoliubov resonances. While values of g≃10 are large compared
with current experimental values of g∼0.5 (see discussion of experimental
g in Rebuzzini ()), resonances at small g∼1 more suitable for
experimental spectroscopy can be excited by considering larger T.
The curve Kϵ=0.2 is in the overlapping
resonance regime, so produces generalised instability.

Figure 1: (a) Shows that for weak kicks (solid line), instability occurs only at one isolated
Bogoliubov (1,1) resonance , where (n,l) denotes the n−th resonance of eigenmode l.
“Up” arrows indicate
onset of exponential instability; “ down” arrows means stability is regained.
g=1. The total number of
non-condensate atoms generated after 200 kicks, Nex(N=200),
is plotted as a function of kicking period T.
For stronger kicks (dotted line; K=1,T<10) resonances
proliferate and there is instability over almost all the parameter range.
(b) Time-dependence near the(1,1) resonance at T≈10
corresponding to Fig (a).
Non-resonant (T=13) curve shows weak quasi-periodic oscillations
in Nex; the near-resonant regime, T=10
is characterized by slow, large oscillations; at resonance T=10.5,
there is exponential growth in Nex(t).
Inset shows that the condensate energy (calculated from the GPE itself) has similar
oscillations.Figure 2: double-kicked BEC (QKR2): Shows zones of instability occur
at Bogoliubov resonances. Condensate losses
as a function of nonlinearity parameter g.
“Up” arrows indicate
onset of exponential instability; “ down” arrows means stability is regained.
Nex(t=1000) is plotted as a function of g (for T=2, ϵ=1/25) for
weak kicks (K=1 so effective kick is Kϵ=0.04) and stronger
kicks (K=5 so effective kick Kϵ=0.2).

In order to understand the behavior at the resonances, we
introduce in Section II a model for the time evolution of perturbations
from the kicked condensate, based on the usual linearization with respect to
small perturbations.

Iv II: Kicked condensate model

The time-evolution of small perturbations of the condensate itself
are described by an equation similar to Eq.4, see castin ().
We write the condensate wavefunction
in the form ψ=ψ0+δψ, where ψ0 is the
unperturbed condensate and δψ represent
the excited components. Inserting this form in the GPE
and linearizing with respect to δψ,
we can write:

iℏddt(δψδψ∗)=L(t)(δψδψ∗).

(5)

where,

L(t)=(H(t)+g|ψ|2gψ∗2−gψ∗2−H(t)−g|ψ|2).

(6)

The analysis of condensate stability for a time-periodic system
Dalfovo () reduces to the analysis of the operator L(t)
over one period T. In general, for systems like BECs in modulated optical lattices,
inter-mode coupling requires a detailed analysis of the instantaneous
evolution. The nature of the
δ-kicked potential permits considerable simplification.

The effect of L(t) reduces to the
free-ringing of the eigenmodes of the unperturbed condensate
for period T, interspersed by instantaneous impulses which mix the modes.
Even for an experiment (where the kicks are approximated pulses of very short, but
finite duration) numerical time-propagation is avoided: intermode
coupling occurs over a very short time-scale, during which eigenmode phases
remain essentially constant.

Excluding the kick term for the moment, we recall that the time propagation
under HGP can be analyzed in terms of the eigenmodes (uk(t),vk(t))
and eigenvalues of ωk(t) of the
2×2 matrix on the right hand side of Eq.6.
Setting ψ=1/√2π,
the matrix can be diagonalized and there are well-known analytical
expressions for the unperturbed eigenmodes Stringari ()

(uk(t=0),vk(t=0))=(UkVk)eikθ√2π,

(7)

where Uk+Vk=Ak,Uk−Vk=A−1k, and
Ak=(ℏ2k22(ℏ2k22+gπ))1/4.

In order to understand the behavior at the resonances, we introduce
below a simple model using the eigenmodes Eq.7 as a basis.
Writing the small perturbation in this basis:

(8)

Neglecting the kick, evolving the modes from some initial time t0,
each eigenmode (uk,vk) simply acquires
a phase ie:

bk(t)=bk(t0)e−iωk(t−t0),

(9)

where ωk=√k22(ℏ2k22+gπ).

After a time interval T, a kick is applied which couples the
eigenmodes. Its effect is obtained by expressing the perturbation in a momentum basis,
ψ=∑lal(t)|l⟩ where
|l⟩=eilθ√2π, and we can restrict ourselves
to the symmetric subspace al=a−l of the initial condensate (parity is
conserved in our system).
Then, we can see by inspection that

ak(t)=Ukbk(t)+Vkb∗−k(t).

(10)

Note that bk=b−k for this system.
Conversely, the corresponding amplitude bk in each eigenmode k is
given trivially from Eq.8 using orthonormality
of the momentum states and the relation U2k−V2k=1, yielding

bk(t)=Ukak(t)−Vka∗k(t).

(11)

If the evolving condensate is given in the momentum basis, the effect
of a kick operator Ukick=e±iKℏcosθ is well-known.
The matrix elements:

Unl=⟨n|Ukick|l⟩=Jn−l(K/ℏ)i±(l−n)

(12)

The Jn−l are Bessel functions.

The amplitudes al(t) are given by

an(t+)=∑li±(l−n)Jn−l(Kℏ)al(t−),

(13)

where an(t+)/al(t−) denotes the amplitude in state |n⟩
just after/before the kick.

We can now define a “time-evolution” operator
L′(T)=B−1Lfree(T)BUkick,
where Lfree denotes free ringing of
the eigenmodes, B is the transformation from momentum
basis to Bogoliubov basis and Ukick is the action of the kick.
A usual procedure for stability analysis of a driven condensate is to
examine the eigenvalues of L′(T) to ascertain whether
there is one (or more eigenvalues) which have a real, positive component
Dalfovo (), ie whether they produce
exponential growth in the amplitudes a±l.

However, to compare with GPE numerics, we simply evolve
the mode amplitudes in time over a few kicks and examine the overall
condensate response to the kicking (in the limit of very weak kicking).
Hence we can evolve the amplitudes al(t=NT) of the condensate
perturbation from period N to period N+1:

a((N+1)T)=L′(T)a(N),

(14)

using only the simple analytical coefficients in Eq.13
and Eq.9,
provided we use the simple transformations in Eqs.10
and Eq.11 to switch between the Bogoliubov mode basis and the momentum basis.
L′(T) is non-unitary,
but the method is quantitative in the perturbative limit
provided ψ≃ψ0, ie we assume a0(N)=a0(0)=1.

We calculate the average energy over the first few N kicks,
⟨E(N)⟩=1N∑Nt=1E(t).
Slow, large amplitude oscillations in E(t) yield a large
⟨E(t)⟩ and indicate
a resonance. Fig.3(a) shows the QKR-BEC behavior,
for equivalent parameters to Fig.1(a).
For low K=0.2, there is the same single (1,1) resonance at T≈10
as in Fig.1(a).
For higher K=1 the method is far from quantitative:
the model Eq.14 is only a valid means of time-evolving
the perturbation over a few kicks for small
K<<1 since it assumes the perturbed component
is negligible; nevertheless, for K=1 it illustrates the regime
of dense, overlapping resonances.

In Fig.4(b) we compare the perturbative Eq.(14) results with full GPE numerics
for the first 20 kick pairs of the QKR2 in the limit of weak kicks.
It shows remarkably good agreement.
Moreover the scaling of the resonances with K is well described.
The QKR2 resonant Bogoliubov spectrum differs appreciably from the
QKR case.
Fig.4(b) shows that for QKR2, even for low K=1,ϵ=1/25
ie Kϵ≈0.04
and low g=1, both l=1 and l=2 resonances are strongly excited.
The QKR2-BEC resonance intensity depends strongly on
K: the l=2 resonances scale as K4, while the l=1 scaling is closer to
K2. In the full GPE numerics, the position of the maxima depends slightly
on K and g, but remains within a few percent of the unperturbed value,
even for longer kicking times if Kϵ remains small.

In the limit of weak driving, one can obtain explicit expressions
for the condensate wavefunction as a function of time.
We assume that a0≈1/√2π≫al≠0.
Then Eq.13 can be approximated by
an(t+)≈al(t−)+Ul0/(√2π).
From Eq.11 and Eq.9 we see that the amplitude
accumulated over a single period in
each eigenmode is

bl(N+1)=bl(N)+(UlUl0−VlU∗l0)eiωlT.

(15)

Summing all contributions iteratively from t=0, taking bl(0)=0, we obtain

bl(N)=(UlUl0−VlU∗l0)n=N−1∑n=0einωlT,

(16)

and so for ωlT≈2π all the contributions add in phase,
analogously to the well-known (but unrelated) resonances of the non-interacting
limit Darcy ().

We can write ∑N−1n=0einωlT=e−i(N−1)ωlTΦ(NωlT2) where the Φ function is:

Φ(NωlT2)=sin(NωlT/2)sin(ωlT/2).

(17)

We thus expect oscillations in each set of ±l momentum components
of amplitude

|2al(N)|2∝4|Ul0|2Φ2(NωlT2).

(18)

Off-resonance there will be quasi-periodic oscillations (in e.g. the condensate energy)
from the superposition of contributions characterized by different
eigenfrequencies ωl.
Close to resonance, a single component dominates; if the l−th mode
is resonant we can write ωlT≈2πM+2δ
where 2δ≪1 is the de-phasing from resonance.
Then

|al(N)|2∝|Ul0|2δ2sin2(Nδ),

(19)

and there are slow, periodic oscillations of large amplitude
∼4|Ul0|2δ2,
at a frequency δ which is not related to any eigenmode frequency,
but given rather by the de-phasing from resonance.

The QKR2 resonant excitation spectrum is rather different
from the QKR, and is analysed further
in the next section.

V III: Resonances of the QKR2-BEC

In the limit Kϵ→0, we can obtain analytical expressions
for the BEC wavefunction of the double-kicked system.
Firstly note that when gϵ≪1,
the non-linearity has little effect during the
short time-interval ϵ. Using the relation,

e+iKℏcosθe−ip2tℏ/2e−iKℏcos(θ)=e−it2ℏ[^p+Ksinθ]2,

(20)

the time evolution can be given as a ‘one-kick’ operator

^U(T)≈U(0)GP(T,0)e−iϵ2ℏ[^p+Ksinθ]2.

(21)

In the limit pϵ≈0, one can split the operators in
Eq.21 and neglect a term Ksinθ^p
to obtain the approximation

^U(T)≈e−i2ℏ^p2T.e−iℏ[K2ϵ2sin2θ−iKϵℏcosθ],

(22)

leaving an effective single-kick quantum rotor with a kicking potential

Vkick=[K2ϵ2sin2θ−iKϵℏcosθ]∑Nδ(t−NT).

(23)

The second term, curiously, appears as kicking potential with an
imaginary, and ℏ dependent, kick strength
iKℏ. It is of purely quantum origin as it arises from
the non-commutativity of p and sinθ, i.e.

iKℏcosθ=[Ksinθ,^p].

(24)

Nevertheless, as seen below, it is important for weak driving as
it controls the amplitude of the first excited mode l=±1.

Figure 3: Average energy ⟨E⟩ after 40 kicks.
The dashed lines indicate the model of Eq.14;
all other plots use full numerics.
The label (n,l) denotes n−th
resonance of mode l.
Resonances of the QKR-BEC for parameters
comparable to Fig.1a. For low g=1, K=0.2, only the single
isolated (1,1) resonance is seen.
For higher K=1, resonances proliferate and overlap.Figure 4: Comparison between full GPE numerics and
the model of Eq.14 for the QKR2-BEC, showing
excellent agreement.
Average energy ⟨E⟩ after 20 kick-pairs.
The label (n,l) denotes n−th
resonance of mode l.
g=1 and ϵ=1/25 so K=1 corresponds to effective
kick strength Kϵ=0.04. For low K, the l=1 resonance amplitudes
scale as ∼K2 while those of the l=2 modes scale as ∼K4.

The matrix elements of the modified kick Vkick, like those in
Eq.13, are Bessel functions. Specifically, the effect of Vkick
on the condensate amplitudes al is given by

an(t+)=∑lUnlal(t−),

(25)

where Unl=∑min−l−mJm(K2ϵ4ℏ)Jn−l−2m(iKϵ), and an(t±) indicates momentum amplitudes
before(-) and after(+) the kick, as in Eq.13.
Since Kϵ≪1 and J|n|>1(z)≃0,
only Bessel functions of low order (m=0 or 1) will be non-negligible,
and we can use the small-argument approximations for them, namely
J0(z)≈1, J±1(z)≈±z/2.

Then, if the condensate is relatively unperturbed, the main effect of the
kick will be to simply excite a
small amount of l=±1 and l=±2 from the |0⟩ state

e−iℏVkickψ≈e−iℏVkick|0⟩=∑lUl0|0⟩

(26)

where

∑lUl0|0⟩

≈

1√2π+iJ1(iKϵ2)|±1⟩+iJ1(K2ϵ4ℏ)|±2⟩

(27)

≈

1√2π−Kϵ4|±1⟩+iK2ϵ8ℏ|±2⟩

We obtain a similar equation to the QKR-BEC for the mode amplitudes, i.e.
bl(N)=(UlUl0−VlU∗l0)∑N−1n=0exp[inωlT].

But if only the lowest excited modes are significant,
then, in particular,
b1(N)=−Kϵ4(U1−V1)∑N−1n=0exp[inω1T]
and
b2(N)=iK2ϵ8ℏ(U2+V2)∑N−1n=0exp[inω2T].
For ωlT≈2π
all the contributions add in phase and we will have a resonance of
either the l=1 or l=2 modes, the regime illustrated in Fig2(b).

Similarly as for the QKR-BEC, we can sum all the contributions to obtain an
approximate analytical expression for the evolving
condensate wavefunction including
excited modes l=±1 and l=±2,

ψ(N)≈12π[1+C1Kϵ2cosθ+C2K2ϵ4ℏcos2θ].

(28)

where

C1

=

−Φ(N~ω1)[cos(N−1)~ω1−iA−21sin(N−1)~ω1],

C2

=

Φ(N~ω2)[A22sin(N−1)~ω2+icos(N−1)~ω2,

and ~ωj=ωjT/2.

Eq.28 shows that the amplitudes |a1|2 and |a2|2
scale as K2 and K4 respectively, as seen in the numerics in
Fig.4(b). Fig.5(a) shows that Eq.28 gives
excellent agreement with GPE numerics,
giving accurately the non-resonant quasi-periodic condensate oscillations.
Near the l=2 resonance of Fig.2, Fig.5(b) confirms
the QKR2 condensate oscillations (obtained from the GPE) scale quite accurately as
∝1(δ)2sin2Nδ as expected from
Eq.19 and Eq.28.

Fig.5(c) shows that, near-resonance, there are
corresponding large oscillations in the non-condensate numbers calculated
from Eq.4.
Near-resonance, Nex increases quadratically with time, on-resonance,
the increase is exponential.

Figure 5: Test of perturbative model.
(a) Condensate energy oscillations
from GPE numerics and Eq.28. Kϵ=0.04, g=2, T=2. Beating between
modes 1 and 2 is very accurately described by Eq.28.
(b) Behavior of l=2 resonance of Fig.1(b) Kϵ=0.04,T=2 and
g≈9.5. As the resonance is approached the amplitude of the
oscillations is proportional to the
square of their wavelength, i.e. E(t=NT)∝K41(δ)2sin2Nδ
where 2δ is the distance from the resonance peak.
(c) Corresponding number of non-condensate atoms
from Eq.4.

Vi IV: Bogoliubov resonances for T=2π

The kick period T=2π, in a non-interacting system of cold atoms
(i.e. g=0) corresponds to a so-called “quantum anti-resonance”
where the cold atom cloud exhibits periodic (period-2) oscillations.
Hence the isolated Bogoliubov resonance regime to higher
K than would be expected for generic T.
The effect of a non-zero g for T=2π was investigated
in Zhang (). An instability border occurring at a critical
value of nonlinearity, e.g. for g≃2 at K=0.8,
was identified where the growth on non-condensate particles with time
became exponential.

In Fig.5(a) we investigate the behavior near critical g,
for K=0.8. We see that if a wider range of g is considered,
the stability border is also a resonance: the condensate rapidly
recovers stability after the instability border is passed.
The condensate is exponentially unstable for g≃2→2.6,
but is quite stable for both g=1.5 and g=3, as shown. Fig.5(b)
shows oscillations in the condensate energy, as a function of time;
a smoothed plot is also shown. For g=1.5 and g=2.8 (off-resonance)
the smoothed plots are flat; for g=2.2 and g=2.5 (near-resonant),
slow deep oscillations are apparent.

The behavior is analogous to that of generic T; however,
the analysis of the condensate resonances for T=2π is less
straightforward: the strongest resonances, even for low K≲2,
do not in fact occur for ωlT≈2πM, where M=1,2,3....

A significant difference between generic T and T=2π is that, for
the generic case, if we write

ωlT≈2πMl+2δ(l)

(29)

we see that for arbitrary generic T, the distance from the
nearest resonance, for the different modes, depends on l.
In contrast, for T=2π, for large l (i.e. l≳3)
we find ωlT≈(l2+gπ)π; in
other words, the de-phasing from the nearest resonance (and hence
the period of the mode oscillations) is
similar (either 2δ≈gπ or
2δ≈1−gπ)
for for all modes. So all mode oscillations for high l
are approximately in phase with each other.

For K=0.8, only low modes l=1,2 are significantly populated.
These low modes (l=1 and l=2) are only in phase with each other
at certain precise values of g,T. For these parameters, the
model of Eq.14 predicts large
resonances whenever the condition (ω1+ω2)T≈2πM
is satisfied. In particular, for the resonance near g≈2,
we find that for the l=1 mode, ω1T≈(1−2δ)2π while
for the l=2 mode ω2T≈(2+2δ)2π,
with 2δ≈.25.

These results suggest that “two-mode resonances”, i.e. synchronized
oscillations of pairs of the lowest excited modes are the
dominant mechanism for T=2π (NB this could be viewed as a “three-mode”
resonance, if we include the lowest, initial mode, but ω0=0 for
our system).
They account for the shifting position of the critical instability
border found by Zhang () in the T=2π case. For example,
for slightly higher kick strengths, such as K≃2, a
resonance appears for g≈1.65 corresponding to
(ω2+ω3)T≈2πM,
which accounts for the displacement of the instability border to lower
values of g. Note that the resonance positions in the full numerics
are K-dependent,
whereas in the perturbative model of Eq.14 this dependence is neglected;
the model is only valid for very small K.

Figure 6: (a) Non-condensate particles for kicking period
T=2π, K=0.8, g≃2. The inset shows
the rate of exponential growth of non-condensate atoms; zero denotes
polynomial growth or less. The graph shows this instability border
is a resonance: the condensate is unstable for g=2−2.5 but is
stable for g=1.5 and g=3.0.
(b) Energy oscillations as a function of
time; smoothed plots are also shown.
Before and after the resonance (g=1.5 and g=2.8) the smoothed
plots are flat. Near-resonance, (g=2.2 and g=2.5) the energy
shows the characteristic slow, deep resonant oscillations.

Vii V: Conclusion

In conclusion,
we have shown that exponential instability in kicked BECs is related
to parametric resonances, ie driving of low-lying collective modes
at their natural frequencies, rather than to
chaos in the underlying mean-field dynamics gard2 ().

The signature of this process
is in the onset of slow, large amplitude periodic oscillations in the
condensate energy as well as the number of non-condensate atoms
calculated from the time-dependent Bogoliubov formalism, as a resonance
is approached. The resonances proliferate and overlap
for large kick-strengths K, leading to instability over wider ranges of
K and g.
The time-dependent Bogoliubov approximation used here and in all
other previous studies is only valid in regimes where the condensate
depletion is negligible; for realistic condensates analysis of the
dynamics in the narrow (for weak driving) windows of parametric instability,
would require other approaches beyond Bogoliubov.
However, away from these windows, the kicked condensate remains
stable and relatively unperturbed, even after prolongued kicking.

JR acknowledges funding from an EPSRC-DHPA scholarship.
The authors would like to thank Chuanwei Zhang for valuable advice.
This research was supported by the EPSRC.

(24)While we can not draw any conclusions on the Kicked Harmonic Oscillator,
as we do not study it,
we note that eg Figs 18 and 19 in the study in [3] show deep slow oscillations
suggestive of an approach to a Bogoliubov resonance (not necessarily
leading to exponential behavior in those examples).